# Tagged Questions

43 views

### Prove by induction a formula for $x_{k+1}=\frac{x_k}{x_k+2}$, $x_1=1$

I have a IT Maths exam coming up and I just can't figure out this question. Any help would be appreciated thanks. A sequence of integers $x_1,x_2,\dots,x_k,\dots$ is defined recursively by ...
19 views

### Clarification regarding the Josephus problem in Concrete Mathematics (Knuth, et al)

In page 9 of Concrete Mathematics, regarding the Josephus Problem, they state that "each person's number has been doubled then decreased by 1". $J(2n) = 2J(n) - 1$, for $n \ge 1$ I don't quite ...
64 views

### Elementary Proof of sum of alternating factorials

so I came across the rather interesting identity $$\sum_{i = 0}^{\infty}\left[\sum_{j=0}^{i}\left[\frac{(-1)^j}{j!} \right] \prod_{j=0}^{i}\left[ (x-j)\right] \right] = x!$$ For positive integers ...
104 views

75 views

### Cayley's formula for the number of trees(Recursion)

I'm trying to understand the proof by recurcion and induction for the Cayley's formula for the number of trees.While I'm trying to understand it there are some things that I don't get at all. -It ...
26 views

### Recurrence relation by expansion

I'm trying to find a general formula for the following recurrence relation: for n of the form 2^2^k S(n) = (rootn)(S(rootn))+n S(2) = 1 First, I let b = 2^2 just for readability ...
62 views

### Got stuck in proving monotonic increasing of a recurrence sequence

I'm stuck with the prove of the following recurrence sequence which was part of and old exam. $a_1:=\frac{1}{2}, a_{n+1}:=a_n(2-a_n)$ for $n \in \mathbb{N}$ I have to show that $0 <a_n < 1$ ...
54 views

### Using induction to prove a general form from a recurrence relation

I have the recurrence relation: $a_{n+2} = \dfrac{-a_{n}}{(n+2)(n+1)}$. I have done some work to identify that two cases emerge: one is n is positive, and one if n is negative. If n = 2m (even) ...
128 views

79 views

### Induction Proof (relating two recurrences)

Let $L(n) = n + 2 L\left(\frac{n}{2}\right), \, L(1) = 1,$ and $U(n) = 9n + 2U\left(\frac{n}{2}\right), \, U(1) = 9.$ Prove by induction that $U(n) = 9L(n)$ where $n = 2^k$. I attempted to prove ...
46 views

### Solve non-linear recurrence

I am trying to solve the following recurrence (i.e. determine $a_n$). I'm not entirely sure how to proceed, I have only encountered linear ones so far. This is not a homework, I'm just doing it for ...
44 views

### Fibonacci Recursion Equation

For the Fibonacci sequence, prove the formula $a^2_{n+1} = a_n a_{n+2} + (-1)^n$ using induction. I have done the base case, when $n=3$, because for the Fibonacci sequence, $a_1=a_2=1$. I have no ...
90 views

### How do you solve a recurrence with a summation function inside

Show that $$t(n) = 1 + \sum_{ j=0}^{n-1} t(j)$$ is the same as $$t(n) = 2^n$$ Initial condition $t(0) = 1$
227 views

### Prove the Number of Additions of Fibonacci Number Algorithm

I am studying for a final exam and I'm having trouble with this question: The following recursive algorithm FIB takes as input an integer $n \ge 0$ and returns the $n$-th Fibonacci number $F_n$: ...
62 views

### Induction proof with no terms of sequence

The sequence $[x_n]$ is given by $x_1=1$ and $x_{n+1}=\displaystyle\frac{4+x_n}{1+x_n}$ for $n\ge 1$. Prove by induction that for $n\ge 1$, ...
595 views

222 views

### $n$ Lines in the Plane

How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines ...
133 views

### Using complete induction, prove that if $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$, then $a_n=2^n$

Could anyone please explain to me how to do this problem by using the principle of complete induction? Thanks. :) Let $a_1=2$, $a_2=4$, and $a_{n+2}=5a_{n+1}-6a_n$ for all $n\geq 1$. Prove that ...
54 views

### Prove by induction that $d_n=2^n+3^n$, where $d_n = 5d_{n-1}-6d_{n-2}$

I have one more induction question. $d_0 =2$ $d_1=5$ let $d_n=5d_{n-1} - 6d_{n-2}$ Prove that $d_n=2^n+3^n$
32 views

### recurrence relation with induction

The following recurrence relation: $T(n)=T(n-1)+n=1+\frac{n^2+n}{2}=\theta(n^2)$, so this mean that: there is $c_1, c_2, n_o > 0 : c_1n^2<=1+\frac{n^2+n}{2}<=c_2n^2$, the second inequality ...
63 views

138 views

### Mathematical induction proof; $g_k=3g_{k-1} - 2g_{k-2}$

Can someone help me with this problem? I'm having a hard time proving this. It's been a long time since I have done mathematical proofs. Suppose that $g_1,\ g_2,\ g_3,\ \ldots$ is a sequence of ...
131 views

### Proving the equality of 2 functions

You are given that $f(n)=g(n, f(n-1))$ (some initial values given). Looking at the first few terms, it becomes obvious that $f(n)=h(n)$. How does one go about proving this? Induction seemed obvious ...